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A group of psychologists developed a test, after which each person gets a mark, the number $Q$, which is the index of his or her mental abilities $($the greater $Q$, the greater the ability$)$. For the country’s rating, the arithmetic mean of the $Q$ values of all of the inhabitants of this country is taken.

a) A group of citizens of country $A$ emigrated to country $B$. Show that both countries could grow in rating.

b) After that, a group of citizens from country $B$ $($including former ex-migrants from $A\,)$ emigrated to country $A$. Is it possible that the ratings of both countries have grown again?

c) A group of citizens from country $A$ emigrated to country $B$, and group of citizens from country $B$ emigrated to country $C$. As a result, each country’s ratings was higher than the original ones. After that, the direction of migration flows changed to the opposite direction – part of the residents of $C$ moved to $B$, and part of the residents of $B$ migrated to $A$. It turned out that as a result, the ratings of all three countries increased again $($compared to those that were after the first move, but before the second$)$. $($This is, in any case, what the news agencies of these countries say$)$. Can this be so $($if so, how, if not, why$)$?

$($It is assumed that during the considered time, the number of citizens $Q$ did not change, no one died and no one was born$)$.

There are $n$ random vectors of the form $(y_1, y_2, y_3)$, where exactly one random coordinate is equal to 1, and the others are equal to 0. They are summed up. A random vector a with coordinates $(Y_1, Y_2, Y_3)$ is obtained.

a) Find the mathematical expectation of a random variable $a^2$.

b) Prove that $|a|\geq \frac{1}{3}$.

A sequence consists of 19 ones and 49 zeros, arranged in a random order. We call the maximal subsequence of the same symbols a “group”. For example, in the sequence 110001001111 there are five groups: two ones, then three zeros, then one one, then two zeros and finally four ones. Find the mathematical expectation of the length of the first group.

On weekdays, the Scattered Scientist goes to work along the circle line on the London Underground from Cannon Street station to Edgware Road station, and in the evening he goes back $($see the diagram$)$.

Entering the station, the Scientist sits down on the first train that arrives. It is known that in both directions the trains run at approximately equal intervals, and along the northern route $($via Farringdon$)$ the train goes from Cannon Street to Edgware Road or back in 17 minutes, and along the southern route $($via St James Park$)$ – 11 minutes. According to an old habit, the scientist always calculates everything. Once he calculated that, from many years of observation:

– the train going counter-clockwise, comes to Edgware Road on average 1 minute 15 seconds after the train going clockwise arrives. The same is true for Cannon Street.

– on a trip from home to work the Scientist spends an average of 1 minute less time than a trip home from work.

Find the mathematical expectation of the interval between trains going in one direction.

At the ball, there were n married couples. In each pair, the husband and wife are of the same height, but there are no two pairs of the same height. The waltz begins, and all those who came to the ball randomly divide into pairs: each gentleman dances with a randomly chosen lady.

Find the mathematical expectation of the random variable X, “the number of gentlemen who are shorter than their partners”.

In a tournament, 100 wrestlers are taking part, all of whom have different strengths. In any fight between two wrestlers, the one who is stronger always wins. In the first round the wrestlers broke into random pairs and fought each other. For the second round, the wrestlers once again broke into random pairs of rivals $($it could be that some pairs will repeat$)$. The prize is given to those who win both matches. Find:

a) the smallest possible number of tournament winners;

b) the mathematical expectation of the number of tournament winners.

At the Antarctic station, there are n polar explorers, all of different ages. With the probability p between each two polar explorers, friendly relations are established, regardless of other sympathies or antipathies. When the winter season ends and it’s time to go home, in each pair of friends the senior gives the younger friend some advice. Find the mathematical expectation of the number of those who did not receive any advice.

In a shopping centre, three machines sell coffee. During the day, the first machine can break down with a probability of 0.4 and the second with a probability of 0.3. Every evening, Mr Ivanov, the mechanic, comes and repairs all of the broken-down coffee machines. One day, Ivanov wrote, in his report, that the mathematical expectation of breakdowns during one week is 12. Prove that Mr Ivanov is exaggerating.

The teacher on probability theory leaned back in his chair and looked at the screen. The list of those who signed up is ready. The total number of people turned out to be n. Only they are not in alphabetical order, but in a random order in which they came to the class.

“We need to sort them alphabetically,” the teacher thought, “I’ll go down in order from the top down, and if necessary I’ll rearrange the student’s name up in a suitable place. Each name should be rearranged no more than once”.

Prove that the mathematical expectation of the number of surnames that you do not have to rearrange is 1 + 1/2 + 1/3 + … + 1/n.

Chess board fields are numbered in rows from top to bottom by the numbers from 1 to 64. 6 rooks are randomly assigned to the board, which do not capture each other $($one of the possible arrangements is shown in the figure$)$. Find the mathematical expectation of the sum of the numbers of fields occupied by the rooks.

A ticket for a train costs 50 pence, and the penalty for a ticketless trip is 450 pence. If the free rider is discovered by the controller, he pays both the penalty and the ticket price. It is known that the controller finds the free rider on average once out of every 10 trips. The free rider got acquainted with the basics of probability theory and decided to adhere to a strategy that gives the mathematical expectation of spending the smallest possible. How should he act: buy a ticket every time, never buy one, or throw a coin to determine whether he should buy a ticket or not?

Two hockey teams of the same strength agreed that they will play until the total score reaches 10.

Find the mathematical expectation of the number of times when there is a draw.

At a conference there were 18 scientists, of which exactly 10 know the eye-popping news. During the break $($coffee break$)$, all scientists are broken up into random pairs, and in each pair, anyone who knows the news, tells this news to another if he did not already know it.

a$)$ Find the probability that after the coffee break, the number of scientists who know the news will be 13.

b$)$ Find the probability that after the coffee break the number of scientists who know the news will be 14.

c$)$ Denote by the letter $X$ the number of scientists who know the eye-popping news after the coffee break. Find the mathematical expectation of $X.$

At a factory known to us, we cut out metal disks with a diameter of 1 m. It is known that a disk with a diameter of exactly 1 m weighs exactly 100 kg. During manufacturing, a measurement error occurs, and therefore the standard deviation of the radius is 10 mm. Engineer Gavin believes that a stack of 100 disks on average will weigh 10,000 kg. By how much is the engineer Gavin wrong?

At the power plant, rectangles that are 2 m long and 1 m wide are produced. The length of the objects is measured by the worker Howard, and the width, irrespective of Howard, is measured by the worker Rachel. The average error is zero for both, but Howard allows a standard measurement error $($standard deviation of length$)$ of 3 mm, and Rachel allows a standard error of 2 mm.

a) Find the mathematical expectation of the area of the resulting rectangle.

b) Find the standard deviation of the area of the resulting rectangle in centimetres squared.

Harry thought of two positive numbers x and y. He wrote down the numbers x + y, x – y, xy and x/y on a board and showed them to Sam, but did not say which number corresponded to which operation.

Prove that Sam can uniquely figure out x and y.

There are 9 street lamps along the road. If one of them does not work but the two next to it are still working, then the road service team is not worried about it. But if two lamps in a row do not work then the road service team immediately changes all non-working lamps. Each lamp does not work independently of the others.

a) Find the probability that the next replacement will include changing 4 lights.

b) Find the mathematical expectation of the number of lamps that will have to be changed on the next replacement.

If one person spends one minute waiting, we will say that one human-minute is spent aimlessly. In the queue at the bank, there are eight people, of which five plan to carry out simple operations, which take 1 minute, and the others plan to carry out long operations, taking 5 minutes. Find:

a) the smallest and largest possible total number of aimlessly spent human-minutes;

b) the mathematical expectation of the number of aimlessly spent human-minutes, provided that customers queue up in a random order.

There is a deck of playing cards on the table $($for example, in a row$)$. On top of each card we put a card from another deck. Some cards may have coincided. Find:

a) the mathematical expectation of the number of cards that coincide;

b) the variance of the number of cards that coincide.

The probability of the birth of twins in Cambria is $p$, and no triplets are born in Cambria.

a) Evaluate the probability that a random Cambrian that one meets on the street is one of a pair of twins?

b) There are three children in a random Cambrian family. What is the probability that among them there is a pair of twins?

c) In Cambrian schools, twins must be enrolled in the same class. In total, there are $N$ first-graders in Cambria.

What is the expectation of the number of pairs of twins among them?